Epistasis, the interaction among genes, is speculated to be ubiquitous in the genetic control of most common human diseases, e.g., obesity, hypertension, and cancer. The animal models have proved to be a powerful approach to understanding genetic architectures and etiologies of common human diseases. The ability to control both genotype and environment in inbred populations of animals greatly simplifies analysis of complex interactions. The statistical modeling of interaction effects among quantitative trait loci (QTL) must accommodate a very large number of potential genetic effects, even when one assumes only a moderate number of QTL. This fundamental statistical challenge presents a major barrier to determining genetic model with respect to the number of QTL, their genomic positions and their genetic effects. The proposed research will develop statistical methodologies and computer software for identifying multiple genes with complex interaction patterns using the Bayesian framework and Markov chain Monte Carlo (MCMC) algorithms. The methods proposed herein will be developed primarily for arbitrary mating designs derived from two inbred lines (e.g., F2, backcrosses, recombinant inbred lines, advanced intercross lines) or multiple inbred lines (e.g., four-way crosses, eight-way crosses). The specific objectives of this proposal are to: (1) establish novel Bayesian model choice and search strategies for identifying epistatic QTL across the entire genome and jointly inferring the number of QTL, their genomic positions and their main and epistatic effects in arbitrary mating designs derived from two inbred lines, (2) develop Bayesian methods and MCMC algorithms for mapping epistatic QTL for complex ordinal traits (e.g., disease susceptibility and severity), and jointly analyzing multivariate continuous and ordinal traits, (3) develop Bayesian methods and MCMC algorithms for mapping epistatic QTL in arbitrary mating designs derived from multiple inbred lines, (4) evaluate the properties of all procedures developed by extensive simulation studies, (5) apply the methods developed to multiple real data sets and compare the proposed methods with some existing methods, and (6) release high quality, user-friendly software to implement the proposed methods. The proposed methods are expected to aid the discovery of a greater number of QTL, improve the accuracy of estimating their genomic positions and their genetic effects, and finally enhance our ability to understand human diseases.